since \(mv^2\)  is the typical energy of ejected particles and \(n v\)  (\(n\)  being the mean density of particles) is the rate at which particles enter the initially empty half-line \(x<0\) . Dimensional analysis, similar to that used in the blast wave problem (see [1]}), allows one to estimate the size of the cascade (equivalently the position of the right-most excited particle). One finds again
\(R \sim \left(\frac{\mathcal {E} t^2}{m n}\right)^\frac{1}{3}\) 
